There are some soundcard-esque devices that can output 384kHz. f.ex. this, but this outputs to a I2S interface, you'd require a I2S DAC and some DIY skills to make something out of this.
http://www.exadevices.com/
EDIT: It seems that they recommend the ESStech ES9018 DAC chip for the 384kHz. Now you just need to find something that has an I2S input and that chip.